Middle school maths, year 3 · unit 2
Expanding and Factorising
What is the area of a rectangle x + 2 wide and x + 3 high? Taken whole it is (x + 2)(x + 3); counted piece by piece it is x² + 5x + 6. The same area written two ways, nothing more. Spreading it out is expanding; packing it back is factorising. We start by counting tiles.
Width × height, spread out
A board to adjustWhat does (x + 2)(x + 3) become when written out? Build the rectangle with the gold handle and count the tiles inside.
Three formulas in pieces
A board to arrangeIs (a + b)² equal to a² + b²? Fit the pieces together and you will see what is missing.
Stack to the same height
A board to adjustCan 2x + 6 be turned into a product? Drag the handle to stack the tiles in several rows.
Tiles back into a rectangle
A board to adjustTurn x² + 5x + 6 back into width × height. Build a rectangle that uses every tile you have.
Try it yourself
Use what you found on the boards. The numbers change every time and the easy questions come first. Get three right and the lamp lights.
What to take away from this unit
- Writing a product out as a sum is expanding; packing a sum back into a product is factorising. Each undoes the other.
- (x + a)(x + b) = x² + (a + b)x + ab. The number in front of x is the sum of the two numbers; the last number is their product.
- (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b². Never drop the 2ab in the middle.
- (a + b)(a − b) = a² − b². A sum times a difference is a difference of squares.
- To factorise, first take out the common factor: ma + mb = m(a + b). Take the largest one so nothing more is left to take.
- For x² + sx + p, find two numbers that add to s and multiply to p and write (x + a)(x + b). If the two numbers are equal it is (x + a)², a perfect square.